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To solve this problem, we need to find the value of \( a \) such that the tangent line to \( y = a^x \) at \( x = 0 \) matches the line \( y = 1 + 3x \).

### Steps:

1. **Find the equation of the tangent line to \( y = a^x \) at \( x = 0 \):**

   - The general form of the tangent line at any point \( x = x_0 \) for a function \( f(x) \) is:
     \[
     y = f(x_0) + f'(x_0)(x - x_0)
     \]
     In this case, we have \( f(x) = a^x \) and \( x_0 = 0 \).

2. **Evaluate \( f(x_0) \) at \( x = 0 \):**

   \[
   f(0) = a^0 = 1
   \]
   So, the value of the function at \( x = 0 \) is 1.

3. **Find the derivative of \( f(x) = a^x \):**

   The derivative of \( a^x \) is:
   \[
   f'(x) = a^x \ln(a)
   \]
   At \( x = 0 \), we get:
   \[
   f'(0) = a^0 \ln(a) = \ln(a)
   \]
   So, the slope of the tangent line at \( x = 0 \) is \( \ln(a) \).

4. **Equation of the tangent line:**

   Using the point \( x = 0 \) and the slope \( \ln(a) \), the equation of the tangent line is:
   \[
   y = 1 + \ln(a)x
   \]

5. **Set the tangent line equal to the given line \( y = 1 + 3x \):**

   We need the slope \( \ln(a) \) to match the slope of the line \( y = 1 + 3x \), which is 3. Therefore, we set:
   \[
   \ln(a) = 3
   \]

6. **Solve for \( a \):**

   To solve for \( a \), exponentiate both sides:
   \[
   a = e^3
   \]

Thus, the value of \( a \) is \( e^3 \).

Would you like further details or have any other questions?

Here are five related questions to expand your understanding:
1. What is the tangent line to \( y = a^x \) at other points \( x_0 \)?
2. How does the value of \( a \) affect the behavior of the function \( y = a^x \)?
3. What would happen if the tangent line equation was different, say \( y = 1 + 2x \)?
4. How do you find the equation of a tangent line for other types of functions?
5. What is the importance of the natural logarithm \( \ln(a) \) in solving problems like this?

**Tip:** The natural logarithm \( \ln(a) \) is crucial in differentiation for exponential functions, especially when solving tangent line problems involving exponentials.

For what value of a is y = 1 + 3x the tangent line for y = a^x at x = 0?

Learn how to find the value of a such that the tangent line to y = a^x at x = 0 is y = 1 + 3x. This solution covers key calculus concepts including derivatives and the natural logarithm. Suitable for high school students in Grades 10-12.

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